Hessian Calculator – Calculator City

Hessian Matrix Calculator

Function f(x, y)

Point (x, y)

What is the Hessian Matrix?

In mathematics, the Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse.

Hessian Matrix Formula

For a function f(x, y), the Hessian matrix is given by:

H(f) = | ∂²f/∂x²   ∂²f/∂x∂y |
       | ∂²f/∂y∂x   ∂²f/∂y² |

The elements of the Hessian matrix are the second-order partial derivatives of the function with respect to the variables x and y.

Practical Examples

Example 1: f(x, y) = x² + y³ at (1, 2)

First, we find the second-order partial derivatives:

∂²f/∂x² = 2
∂²f/∂y² = 6y
∂²f/∂x∂y = 0
∂²f/∂y∂x = 0

At the point (1, 2), the Hessian matrix is:

H(f)(1,2) = | 2   0 |
            | 0  12 |

Example 2: f(x, y) = x²y at (2, 3)

First, we find the second-order partial derivatives:

∂²f/∂x² = 2y
∂²f/∂y² = 0
∂²f/∂x∂y = 2x
∂²f/∂y∂x = 2x

At the point (2, 3), the Hessian matrix is:

H(f)(2,3) = | 6   4 |
            | 4   0 |

How to Use This Hessian Calculator

Enter the function f(x, y) in the “Function f(x, y)” field.
Enter the point (x, y) at which to evaluate the Hessian matrix in the “Point (x, y)” field.
Click the “Calculate” button.
The calculator will display the Hessian matrix at the given point.

Key Factors That Affect the Hessian Matrix

The function itself: The form of the function determines its curvature and, therefore, the values in the Hessian matrix.
The point of evaluation: The Hessian matrix can change depending on the point at which it is evaluated.
The number of variables: For a function with n variables, the Hessian matrix will be an n x n matrix.
The coordinate system: The values in the Hessian matrix can change if the coordinate system is changed.
The scale of the variables: The values in the Hessian matrix can change if the variables are scaled.
The presence of symmetries: If the function has certain symmetries, the Hessian matrix may also have certain symmetries.

FAQ

What is the Hessian matrix used for?

The Hessian matrix is used in a variety of applications, including: optimization, machine learning, and physics.

What is the determinant of the Hessian matrix?

The determinant of the Hessian matrix is called the Hessian determinant. It is used in optimization to determine whether a critical point is a local maximum, local minimum, or saddle point.

What is the relationship between the Hessian matrix and the Jacobian matrix?

The Hessian matrix is the Jacobian matrix of the gradient of the function.

What is the difference between the Hessian matrix and the gradient?

The gradient is a vector of first-order partial derivatives, while the Hessian matrix is a matrix of second-order partial derivatives.

Can the Hessian matrix be non-symmetric?

If the second partial derivatives are continuous, then the Hessian matrix is symmetric.

What is a positive definite Hessian matrix?

A positive definite Hessian matrix indicates that the function is convex at that point.

What is a negative definite Hessian matrix?

A negative definite Hessian matrix indicates that the function is concave at that point.

What is an indefinite Hessian matrix?

An indefinite Hessian matrix indicates that the function has a saddle point at that point.